sum of the divisors of n, is usually written σ(n). If s > 0, Setting s = 1 gives If the Riemann hypothesis is true, and d(n) d(n) = σ0(n) is the number of divisors of n, including 1 and n itself. where γ = 0.5772... is the Euler–Mascheroni constant. φ(n) Euler's totient function φ(n) is the number of positive integers less than n and coprime to n. Ramanujan defines a generalization of it, if is the prime factorization of n, and s is a complex number, let so that φ1(n) = φ(n) is Euler's function. He proves that and uses this to show that Letting s = 1, Note that the constant is the inverse of the one in the formula for σ(n). Λ(n) Von Mangoldt's function unless n = pk is a power of a prime number, in which case it is the natural logarithm log p. Zero For all n > 0, This is equivalent to the prime number theorem. r2s(n) (sums of squares) r2s(n) is the number of way of representing n as the sum of 2s squares, counting different orders and signs as different (e.g., r2(13) = 8, as 13 = (±2)2 + (±3)2 = (±3)2 + (±2)2.) Ramanujan defines a function δ2s(n) and references a paper in which he proved that r2s(n) = δ2s(n) for s = 1, 2, 3, and 4. For s > 4 he shows that δ2s(n) is a good approximation to r2s(n). s = 1 has a special formula: In the following formulas the signs repeat with a period of 4. and therefore, (sums of triangles) is the number of ways n can be represented as the sum of 2s triangular numbers (i.e. the numbers 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15, ...; the n-th triangular number is given by the formula n(n + 1)/2.) The analysis here is similar to that for squares. Ramanujan refers to the same paper as he did for the squares, where he showed that there is a function such that for s = 1, 2, 3, and 4, and that for s > 4, is a good approximation to Again, s = 1 requires a special formula: If s is a multiple of 4, Therefore, Sums Let Then for , See also Gaussian period Kloosterman sum Notes References